Gravity Probe B Gyroscope Lab

Two small rotations of a gyroscope axis in orbit

General relativity predicts that a gyroscope in free fall around the Earth slowly changes direction relative to distant stars. To first post-Newtonian order its spin S obeys dS/dt = Ω × S with

Ω = (3/2) (GM / c² r³) (r × v) + (G / c² r³) [ 3 (J·r̂) r̂ - J ].

The first term is the geodetic (de Sitter) effect: the gyroscope is carried through the curved space around the Earth. For a circular orbit it is constant, along the orbit normal, and equals (3/2)(GM)3/2/(c² a5/2), about 6600 milliarcseconds per year (mas/yr) at 642 km. The second term is the frame-dragging (Lense-Thirring) effect: the spinning Earth (angular momentum J = I ω) drags local inertial frames like a gravitomagnetic dipole. Averaged over an orbit, ⟨ΩLT⟩ = (G / c² a³) [ J/2 - (3/2)(J·n̂) n̂ ]: about 40 mas/yr for a polar orbit, and reversed in sign and doubled for an equatorial orbit.

What the simulation does

It integrates the spin direction with an exact rotation per step (1/60 of an orbit) along a circular orbit, using the full instantaneous Ω(t), and records the drift in two directions: u1 (the direction the geodetic rotation moves the axis) and u2 (cross track). The three in-page checks compare the numerically orbit-averaged formulas with the closed forms and with the published Gravity Probe B predictions.

Honest simplifications

• Circular two-body orbit; no oblateness (J2), drag, perturbations from the Moon and Sun, no proper motion of the guide star, no gyroscope torques or rotor patch effects.
• The Earth is a point gravitomagnetic dipole with I = 8.04e37 kg m²; the real mass distribution changes the frame-dragging value by a few percent (this model: about 41 mas/yr, published prediction 39.2).
• Real rates are millionths of a degree per year, so the 3D drawing multiplies the drift angle by the chosen exaggeration factor; the readouts and graphs show the true angles. The spacecraft is drawn far larger than to scale.
• Gravity Probe B measured 6602 +/- 18 mas/yr and 37.2 +/- 7.2 mas/yr (2011); this page does not reproduce its data analysis.

• Load the Gravity Probe B preset, press the one-year button and compare the final drift with the predicted rates.
• Switch frame dragging off and then geodetic off to see each effect alone; they point in different directions (along u1 and u2).
• Choose the equatorial orbit: the frame-dragging rate changes sign and doubles.
• Raise the altitude to geostationary height and read how fast both effects fall with distance.